Geodesic
Almost proper GIT-stacks and discriminant avoidance
Documenta mathematica, Tome 15 (2010), pp. 957-972
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We prove that the classifying stack of an reductive group scheme over a field is very close to being proper. Using this we prove a result about isotrivial families of varieties. Fix a polarized variety with reductive automorphism group. To prove that every isotrivial family with this fibre has a rational section it suffices to prove this when the base is projective, i.e., the discriminant of the family is empty. 
@article{10_4171_dm_319,
     author = {Jason Starr and Johan de Jong},
     title = {Almost proper {GIT-stacks} and discriminant avoidance},
     journal = {Documenta mathematica},
     pages = {957--972},
     year = {2010},
     volume = {15},
     doi = {10.4171/dm/319},
     url = {http://geodesic.mathdoc.fr/articles/10.4171/dm/319/}
}
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Jason Starr; Johan de Jong. Almost proper GIT-stacks and discriminant avoidance. Documenta mathematica, Tome 15 (2010), pp. 957-972. doi: 10.4171/dm/319

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